Problem 10 - Algebra 1: Concepts and Skills

Question

Are the graphs of y = x + 2 and y = x - 4 parallel lines?

Step-by-Step Solution

Answer

Yes, the graphs of y = x + 2 and y = x - 4 are parallel lines.

Step 1: Identify the slopes of the lines

The slope of a line in the y = mx + b form, where m is the slope and b is the y-intercept, is the coefficient of x. In the equations y = x + 2 and y = x - 4, the slopes of the lines are both 1 since the coefficients of x are 1 in both equations.

Step 2: Compare the slopes

Now that the slopes of both lines have been identified, they can be compared. Both slopes are equal to 1.

Step 3: Conclusion

Since the two slopes are equal and lines are parallel if and only if their slopes are equal, the conclusion is that the graphs of y = x + 2 and y = x - 4 are indeed parallel lines.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, and they describe a straight line on a graph. They are typically written in the form of \( y = mx + b \). In this equation:

• \( y \) represents the dependent variable or the value we are solving for.
• \( x \) is the independent variable, or the input value.
• \( m \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, which shows where the line crosses the y-axis.

Every linear equation can be graphed as a straight line, and each point on that line is a solution to the equation. This format allows us to predict and calculate the value of \( y \) given any \( x \). Understanding linear equations helps in determining the relationship between two variables and is widely used in different fields, ranging from budgeting to physics.

Slope

The slope of a line is a measure of its steepness and is often symbolized by the letter \( m \) in the equation form \( y = mx + b \). The slope is calculated as the "rise over run," which means it represents the change in \( y \) divided by the change in \( x \). This can be calculated using the formula:\[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\]

• Positive slope: The line ascends from left to right.
• Negative slope: The line descends from left to right.
• Zero slope: The line is horizontal.
• Undefined slope: The line is vertical.

For the equations \( y = x + 2 \) and \( y = x - 4 \), both have a slope of 1, indicating that the lines rise equally as they move from left to right. A consistent slope of 1 means that for every single unit increase in \( x \), \( y \) also increases by one unit, resulting in parallel lines.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis on a graph. It is represented by the \( b \) in the linear equation \( y = mx + b \).

• In the equation \( y = x + 2 \), the y-intercept is 2. This means that when \( x = 0 \), \( y = 2 \), and the line crosses the y-axis at the point (0, 2).
• Similarly, for the equation \( y = x - 4 \), the y-intercept is -4. Thus, the line intersects the y-axis at the point (0, -4).

The y-intercept is essential for graphing because it provides the starting point of the line on the y-axis. When graphing a line, knowing its y-intercept allows you to accurately place it on the graph, after which you can use the slope to determine the direction and steepness of the line.